"Vertically and crosswise."
23×14 → vertical: 3×4=12, cross: 2×4+3×1=11, vertical: 2×1=2 → 322
General multiplication of any two numbers. The most versatile Vedic multiplication method.
Multiply vertically and crosswise to get partial products, then combine them for the final answer. Works for any pair of numbers.
This is the general multiplication formula of Vedic Mathematics and can multiply any two numbers regardless of size. For two 2-digit numbers (ab × cd): Step 1 (vertical right): b×d gives the units. Step 2 (crosswise): a×d + b×c gives the tens. Step 3 (vertical left): a×c gives the hundreds. Carries propagate left as usual. The method extends naturally to 3-digit, 4-digit and larger numbers. For 3-digit numbers you get 5 partial products. The pattern is systematic and elegant — each "column" of the answer comes from a specific crosswise pattern. This method is significantly faster than long multiplication because all operations happen in parallel columns.
Crosswise multiplication has roots in ancient Indian mathematical texts. The Lilavati (1150 CE) by Bhaskaracharya describes similar column-based multiplication. However, the systematic formulation as "Urdhva-Tiryagbhyam" is attributed to Bharati Krishna Tirtha's reconstruction. The technique bears remarkable similarity to the lattice multiplication method found in the works of al-Khwarizmi (780–850 CE) and later adopted in medieval European mathematics. Indian merchants used this method for centuries in trade calculations along the Silk Road. The crosswise pattern also appears in Chinese rod-calculus multiplication from the Han dynasty.
The method is a direct implementation of polynomial multiplication. If we represent ab as (10a+b) and cd as (10c+d), then (10a+b)(10c+d) = 100ac + 10(ad+bc) + bd — which is exactly the vertical-crosswise pattern. For n-digit numbers, this generalizes to convolution of digit sequences, the same operation used in digital signal processing (DSP). The Fast Fourier Transform (FFT) is essentially a highly optimized version of this crosswise principle. The Karatsuba algorithm (1960), which was a breakthrough in computational complexity, is also a direct descendant of this divide-and-conquer multiplication approach.